Weak limits for quantum walks on the half-line

TL;DR

This paper establishes a weak limit theorem for quantum walks on the half-line, revealing boundary effects, localization possibilities, and specific behaviors for the Hadamard walk, advancing understanding of quantum walk asymptotics.

Contribution

It formulates and proves a general weak limit theorem for quantum walks on the half-line with boundary conditions, including localization phenomena and specific results for the Hadamard walk.

Findings

Localization can occur even with homogeneity.

The weak limit for the Hadamard walk is independent of initial coin state.

The limit exhibits no localization for the Hadamard walk.

Abstract

For a discrete two-state quantum walk (QW) on the half-line with a general condition at the boundary, we formulate and prove a weak limit theorem describing the terminal behavior of its transition probabilities. In this context, localization is possible even for a walk predicated on the assumption of homogeneity. For the Hadamard walk on the half-line, the weak limit is shown to be independent of the initial coin state and to exhibit no localization.